US 7120601 B2 Abstract A method, system and medium for optimally allocating investment assets for a given investor within and between annuitized assets and non-annuitized assets retrieves an investor's utility of consumption, utility of bequest, objective and subjective probabilities of survival and expected rates of return from each of a plurality of annuity and nonannuity assets having varying degrees of risk and return. Based on these inputs, an objective utility function is maximized by adjusting the asset allocation weights. The optimal asset allocation weights may be used to allocate the assets of the investor's portfolio among predetermined investment vehicles or as an analytical tool by portfolio managers.
Claims(95) 1. A method for allocating assets of an investor portfolio among annuity and nonannuity assets, comprising the steps of:
retrieving at least one probability of survival of the investor;
selecting a utility of consumption and a utility of bequest of the investor;
retrieving for each of a plurality of nonannuity assets, an expected rate of return, the nonannuity assets having expected rates of return which are different from each other;
retrieving, for each of a plurality of annuity assets, an expected rate of return, the annuity assets having expected rates of return which are different from each other;
maximizing by a computer an objective utility function as the sum of a utility of a live state and a utility of a dead state given the retrieved rates of return by adjusting the values of a plurality of investment weighting factors each corresponding to a nonannuity asset or an annuity asset, said maximizing step performed in multiple iterations; and
allocating assets in the portfolio to the nonannuity assets and the annuity assets according to receptive ones of the investment weighting factors.
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calculating each of the investment weighting factors as components of the utility of the live state; and
calculating each of the investment weighting factors of the nonannuity assets as components of the utility of the dead state.
8. The method of
retrieving a subjective probability of survival for the investor; and
performing said step of maximizing by apportioning the objective function between the utility of the live state and the utility of the dead state according to the value of the subjective probability of survival.
9. The method of
retrieving an objective probability of survival for the investor; and
calculating the utility of the live state as a function of the objective probability of survival of the investor.
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(i) the investor's risk tolerance; and
(ii) the investor's age.
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24. A system for allocating portfolio assets of an investor among annuity and nonannuity vehicles, comprising:
a memory for storing at least one probability of survival of the investor, at least one of a utility of consumption attributed to the investor and a utility of bequest attributed to the investor, data identifying a plurality of nonannuity assets having different rates of return and degrees of risk, data identifying a plurality of annuity assets having different rates of return and degrees of risk;
a processor coupled to the memory to retrieve the stored probability of survival, the stored utility of bequest or utility of consumption and said data identifying the annuity and nonannuity assets, the processor maximizing an objective utility function as the sum of a utility of a live state and a utility of a dead state given the retrieved rates of return by adjusting the values of a plurality of investment weighting factors, each factor being a weight of a respective nonannuity or annuity asset, where the processor maximizes the objective utility function by performing multiple iterations of calculation steps; and
a portfolio asset allocator coupled to an output of the processor for allocating assets in the portfolio responsive to the adjusted values of the investment weighting factors which result in a maximum of the objective utility function.
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35. A system for optimally allocating investment assets for a given user within and between annuitized assets and non-annuitized assets, comprising:
a computer having a CPU and memory;
means for providing to the computer at least one of the user's utility of consumption and utility of bequest;
means for providing to the computer the user's objective probability of survival based on the user's age;
means for providing to the computer the user's subjective probability of survival based on the user's physical well-being;
means for providing to the computer an expected rate of return from each of a plurality of assets, ones of the assets being riskier than others of the assets; and
means for maximizing an objective utility function which is calculated as a function of the asset expected rates of return, a utility of a live state of the user and a utility of a dead state of the user and a plurality of asset allocation weights respectively corresponding to the different assets, where the means for maximizing maximizes the objective utility function by performing multiple iterations of calculation steps.
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(i) the investor's risk tolerance; and
(ii) the investor's age.
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41. A machine-readable medium on which has been prerecorded a computer program which, when executed by a processor, performs the following steps:
retrieving at least one probability of survival of an investor;
selecting a utility of consumption and a utility of bequest of the investor;
retrieving, for each of a plurality of annuity assets and nonannuity assets, an expected rate of return, each nonannuity asset and each annuity asset having an expected rate of return and a degree of risk which is different from others of the assets;
maximizing an objective utility function as the sum of a utility of a live state and a utility of a dead state given the retrieved rates of return by adjusting the values of a plurality of investment weighting factors corresponding to respective ones of the annuity and nonannuity assets, where the processor performs said step of maximizing in multiple iterations; and
allocating assets in a portfolio of the investor comprising the nonannuity assets and the annuity assets according to the investment weighting factors for which the objective utility function is maximized.
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calculating the investment weighting factors of the annuity assets and the nonannuity assets as components of the utility of the live state; and
calculating the investment weighting factors of the nonannuity assets as components of the utility of the dead state.
47. The medium of
retrieving a subjective probability of survival for the investor; and
performing said step of maximizing by apportioning the objective utility function between the utility of the live state and the utility of the dead state according to the value of the subjective probability of survival.
48. The medium of
retrieving an objective probability of survival for the investor; and
calculating the utility of the live state as a function of the objective probability of survival of the investor.
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57. A method for allocating assets of an investor to a plurality of annuity and nonannuity investment vehicles, comprising the steps of:
retrieving personal characteristics of the investor, the characteristics including age, a risk aversion measure and at least one of a utility of consumption and a utility of bequest;
retrieving data about the financial assets of the investor, including, for each investment vehicle presently employed by the investor, a measure of risk of the investment vehicle, a rate of expected return for the investment vehicle, and an amount of assets in the investment vehicle, the sum of the assets in the investment vehicles being the total financial assets of the investor;
classifying the investment vehicles of the investor into allocatable investment vehicles whose assets may be reallocated, and nonallocatable investment vehicles whose assets may not be reallocated;
providing a plurality of asset classes having varying degrees of risk and expected return, the asset classes including nonannuity asset classes and annuity asset classes;
attributing the present investment vehicles of the investor to the asset classes as a function of the risk and expected return of the asset class and the investment vehicle;
maximizing an objective utility function as the sum of a utility of a live state and a utility of a dead state given the rates of expected return of the asset classes by adjusting values of a plurality of investment weighting factors each corresponding to an asset class, where said step of maximizing is performed in multiple iterations; and
rebalancing the total financial assets of the investor among the asset classes as a function of the investment weighting factors by adjusting the amount of assets held in each of the allocatable investment vehicles.
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calculating each of the investment weighting factors as components of the utility of the live state; and
calculating each of the investment weighting factors of the nonannuity assets as components of the utility of the dead state.
63. The method of
retrieving a subjective probability of survival for the investor; and
performing said step of maximizing by apportioning the objective function between the utility of the live state and the utility of the dead state according to the value of the subjective probability of survival.
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retrieving an objective probability of survival for the investor; and
calculating the utility of the live state as a function of the objective probability of survival of the investor.
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(i) the investor's risk tolerance; and
(ii) the investor's age.
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75. A method of comparing a current allocation of assets in an investment portfolio by a portfolio manager with an optimum allocation of the assets of the portfolio among a plurality of annuity and nonannuity assets, comprising the steps of:
retrieving, for the investor for which the portfolio has been established, at least one probability of survival;
retrieving for the investor a utility of consumption and a utility of bequest;
retrieving, for each of the annuity and nonannuity assets, an expected rate of return, ones of the annuity assets having rates of return which are different from others of the annuity assets, ones of the nonannuity assets having rates of return which are different from others of the nonannuity assets;
relating a utility of a live state of the investor to a utility of a dead state of the investor as a function of the utility of consumption and the utility of bequest;
maximizing an objective utility function as the sum of the utility of the live state and the utility of the dead state given the retrieved rates of return by adjusting the values of a plurality of investment weighting factors each corresponding to one of the annuity assets or one of the nonannuity assets, where said step of maximizing is performed in multiple iterations;
making a proposed allocation of the assets of the portfolio among the nonannuity assets and the annuity assets according to the relative values of the investment weighting factors;
comparing the proposed allocation to the current allocation; and
modifying the current allocation in view of the proposed allocation.
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calculating each of the investment weighting factors as components of the utility of the live state; and
calculating each of the investment weighting factors of the nonannuity assets as components of the utility of the dead state.
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retrieving a subjective probability of survival for the investor; and
performing said step of maximizing by apportioning the objective function between the utility of the live state and the utility of the dead state according to the value of the subjective probability of survival.
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(i) the investor's risk tolerance; and
(ii) the investor's age.
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94. A method for allocating assets of an investor portfolio among annuity and nonannuity assets, comprising the steps of:
retrieving at least one probability of survival of the investor;
selecting a utility of consumption and a utility of bequest of the investor;
retrieving for at least one nonannuity asset or at least one annuity asset or both, an expected rate of return;
maximizing an objective utility function as the sum of a utility of a live state and a utility of a dead state given the retrieved rate(s) of return by adjusting the values of a plurality of investment weighting factors each corresponding to an asset, where the step of maximizing is performed in multiple iterations; and
allocating assets in the portfolio to the asset(s) according to receptive ones of the investment weighting factors.
95. The method of
Description The present invention relates to a method and system for allocating retirement savings to finance retirement consumption. More particularly, the present invention relates to a method and system for optimally allocating investment assets within and between annuitized assets and non-annuitized assets having different degrees of risk and return. With the large bulk of baby boomers getting closer to their golden years, a great deal of public interest is shifting toward how individuals should finance their retirement. Although a substantial portion of retirement needs is usually provided by company and government pensions, there is often a large gap that can only be funded with discretionary savings. Indeed, based on the Survey of Consumer Finances, the number one reason for households to save is to finance retirement. One of the most important decisions investors have to make is how to allocate their retirement savings to finance their retirement consumption. Investors face two major risk factors when making asset allocation decisions in retirement: financial market risk and longevity risk. Financial market risk is the performance uncertainty of investment vehicles. It is typically measured and quantified by the volatility of the investment returns. Longevity risk, on the other hand, is the uncertainty of how long the consumer will need retirement income (i.e., the risk that the consumer could potentially outlive his or her investments). Financial market risk is related to financial market behavior and economic the environment; longevity risk is normally related to the health condition of the investor. Traditionally, asset allocation is determined by constructing efficient portfolios for various risk levels based on modern portfolio theory (MPT) developed by Harry Markowitz (1952) “Portfolio Selection,” In “Optimum Consumption and Portfolio Rules in a Continuous-Time Model”, Annuitization involves paying a non-refundable lump sum to an insurance company. In exchange, the insurance company guarantees a constant life-long payment stream that can not be outlived, but is terminated upon death. The generic life payout annuity provides no bequest for heirs, although spousal protection, in the form of joint-and-last-survivor, can be purchased at additional cost. Annuities can be viewed as longevity insurance, a type of mortality-contingent claim. To create the contract, the annuity (insurance) company receives money from each person who buys a payout annuity. The funds received from each person are pooled into a collective pool from which claim payments are made. An insured person's annuity claim is contingent on continued life. The annuity payments are calculated so that each recipient gets the largest amount that the company can pay without exceeding the fund created from the accumulations of the entire group. The funds from people who die earlier than average support those who live longer than average. Many people purchase life annuities in order to be assured of an income they can't outlive. Others hesitate to select a lifetime payout annuity because they must give up control of the funds and hence the amount they can pass to their heirs. Just like other types of insurance, the amount and kind of immediate annuities an investor should own depends on many factors, including: (i) how risk averse he or she is; (ii) how much wealth has been accumulated; (iii) how much is the retirement need; (iv) how long he or she needs the savings to last; and (v) how strongly he or she feels about bequest. The fees and costs of annuity investments relative to other investment vehicles (such as mutual funds) as well as the tax structure and the investor's tax bracket also have impacts on the optimal allocations. Despite the importance of this decision to each and every retiree, there have been relatively few scholarly papers written or models developed on the normative topic of the optimal allocation to immediate annuities. Most of the literature tends to focus on a positive economic analysis of the immediate annuity market. For example, a substantial body of literature has documented the extremely low levels of voluntary annuitization exhibited amongst elderly retirees. Strictly speaking, this phenomenon is inconsistent with results of the life-cycle model of savings and consumption disclosed by M. Yaari in “Uncertain Lifetime, Life Insurance and the Theory of the Consumer”, Nevertheless, despite the highly appealing arguments in favor of annuitization, there is little evidence that retirees are voluntarily embracing this arrangement. Notably, very few people consciously choose to annuitize their marketable wealth. In the comprehensive Health and Retirement Survey (HRS), conducted in the U.S, only 1.57% of the HRS respondents reported annuity income. Likewise, only 8.0% of respondents with a defined contribution pension plan selected an annuity payout. The U.S. based Society of Actuaries, conducted a study that shows that less than 1% of variable annuity (VA) contracts were annuitized during the 1992–1994 period. Brugiavini discloses a conventional model in “Uncertainty Resolution and the Timing of Annuity Purchases”, In related research, S. Kapur and J. M. Orszag introduced immediate annuities into a Merton (1971) framework in “A Portfolio Approach to Investment and Annuitization During Retirement”, One-Period Model A life payout annuity is a financial investment whose returns are enhanced by pooling mortality risk with others. Here is a simple example of a one-period life annuity that illustrates the concept of risk pooling. According to U.S. life tables compiled by the Center for Disease Control and Prevention ( Now, imagine that five such 95 year-old females entered into the following legally binding agreement. Each of the five females has agreed to contribute $100 to a communal fund that will invest in Treasury bills yielding 5%. Then, according to the contract, at the end of the year, only the surviving females will be entitled to split the proceeds of the fund. Clearly, the total contribution of $500 will grow to $525 by the end of the year. And, if all five females are still alive—they are now 96 years old—they will each receive $105. This is precisely the $100 investment, plus interest. If one of them dies during the next year, the remaining four will be entitled to split the $525, giving each a total of $131.25. Recall that the agreement stipulated that those who die cede control of their assets. The four survivors will therefore gain a return of 31.25% on their money. In fact, if two happen to die, the remaining three will get $175, which is an impressive 75% return on their money. In other words, the survivors' gains are comprised of their original principal, their interest, and other people's principal and interest. By pooling mortality risk and ceding bequests, everyone gains. Technically, this agreement is called a tontine, also known as a pure endowment contract which will be referred to hereinafter as a one-period life annuity contract. Of course, with only five females in the mortality pool, six different things can happen. In the two extreme cases, they all might die, or they all might survive. But with 10,000 such females entering into a one-period annuity agreement, it is pretty much assured that the $1,050,000 will be split among 8,000 survivors. In other words, the expected return from the contract for the survivors is (1,050,000/8,000)=$131.25, or a 31.25% gain. The numerator is the total return for the pool, and the denominator represents the survivors. Algebraically, if R denotes the risk-free interest rate (U.S. Treasury Bills) per period, and p is the probability of survival per period, then the return from the one-period annuity is (1+R)/p−1>R, where p denotes the probability of survival. This is the return for the survivors. And again, the reason the return is greater than R is because the dead subsidize the living. Furthermore, the smaller the p, the greater is the 1/p and the greater is the return from the one-period life annuity. Now, this arrangement is not as outrageous or artificial as it may first sound. In fact, it is the principle underlying all immediate annuities, and all pension plans for that matter. In practice, the agreement is made over a series of periods, as opposed to just one. But the mechanics are the same, and the survivors derive a higher return compared to placing their funds in a conventional asset (non-mortality contingent) asset. Although the example we have provided assumes that R (the risk-free interest rate) is fixed, the same would apply for a Variable and unknown return denoted by X. Moreover, the exact same principle would apply with a variable investment return as well. In fact, the returns might be even higher. Namely, the 10,000 females can invest their $100 in a stock mutual fund that earns 5%, or 10% or even 20%. They do not know in advance, what the fund/pool will earn. At the end of the year, the annuitants will learn (or realize) their investment returns, and then split the gains among the surviving pool. Moreover, in the event the investment earns a negative return, i.e., loses money, the participants will share in the losses as well, but the effect will be mitigated by the mortality credits. If five females invest $100 in a fund that earns a random return, the expected return for the survivors is (1+X)/p−1>X. This concept is the foundation of a variable immediate annuity, which is the counterpart to a fixed immediate annuity. Both fixed and variable immediate annuities provide longevity insurance via the mortality credits but with fixed annuities the pool has been invested in fixed income securities, and with variable annuities the pool has been invested in variable return (read stocks, real estate, etc.) securities. The choices and decisions made between fixed and variable immediate annuities are identical to the choices between fixed and variable accumulation products. The optimal allocations should depend on the participant's risk aversion, comfort with the fluctuating stock market, time horizon, and budgeting requirements. As such, the fixed immediate annuity is one particular asset class within the portfolio of longevity insurance products. Thus, providing a fixed immediate annuity without access to a variable immediate annuity is akin to offering a money market and bond fund in a (pension) savings plan without providing equity-based products to span the risk and return spectrum. In practice, only insurance-chartered companies are authorized to provide these mortality-contingent products. In fact, most insurance companies go one step further and actually guarantee that you will receive the mortality credit enhancements, even if the mortality experience of the participants is better than expected. In other words, in the above-mentioned example, with an expected 20% mortality rate, they would guarantee that all survivors receive 31.25% on their money, regardless of whether or not 20% of the group died during the year. Insurance companies are able to provide this guarantee by making very careful and conservative assumptions about the rate of return earned on assets. Furthermore, the greater the number of insurance annuities an insurance company sells, or has on its books, the lower the risk is in providing this guarantee. These are the ultimate economics of scale. In other words, the risks might be significant if they only sold five such policies, but with 500,000 policies, the probability of an adverse outcome is close to zero. One aspect of the invention provides a system, method and machine-readable medium for allocating assets of an investor portfolio among annuity and nonannuity assets. According to this method, at least one probability of survival of the investor is retrieved. The investor selects, or has selected for him or her, a utility of consumption and a utility of bequest. Characteristics of a plurality of nonannuity assets and a plurality of annuity assets are retrieved, these characteristics including for each kind of asset at least one differentiating characteristic or moment, such as risk (standard deviation of return), an expected rate of return (based on a lognormal or other random distribution) or other measurable differentiating characteristic (such as skewness and kurtosis). In a preferred embodiment the method maximizes an objective utility function which is calculated as the sum of a utility of live state and a utility of a dead state, and takes into account the relative risk aversion of the investor. The live state and dead state utilities in turn are calculated as functions of the retrieved expected rates of return and values of a plurality of investment weighting factors, each weighting factor being attributed to a respective one of the annuity or nonannuity assets. Once a maximum of the objective utility function has been found, the investment weighting factors that result in that maximum can be used in allocating assets in the investor's portfolio or alternatively can be used by a portfolio manager to compare present allocations with optimum allocations. Preferably, the chosen objective utility function should account for the following factors: (i) the investor's risk tolerance; (ii) the investor's age; (iii) the investor's subjective and objective probabilities of survival; (iv) the investor's utility of consumption, and his or her utility of bequest; (v) the expenses and fees of annuity and nonannuity assets, and (vi) risk and return characteristics of each of the assets to which a portion of the investor's wealth could be allocated. In a preferred embodiment, two different probabilities of survival of the investor are retrieved, an objective probability of survival which is actuarially determined on the basis of age, gender and possibly ethnic origin and a subjective probability of survival which depends upon the investor's own perception of health, genetic factors, etc. The subjective probability of survival is used to apportion the investor's wealth between the utility of the live state and the utility of the dead state. The objective probability of survival is used to determine effective rates of return for each of the annuity assets. Various optimization methods can be used to find a maximum of the objective utility function. In several embodiments, all but one of the investment weighting factors each supplies an additional dimension to a search space, so that a space is created with (n−1) dimensions if there are n investment weighting factors. The last investment weighting factor is found by subtracting the other weighting factors from unity. The (n−1) dimensional search space is subdivided using any one of several known search algorithms, including a Golden Mean search, a Fibonacci search, the Method of Feasible Directions, simulated annealing or a basic “decimal” search in which the space is divided into a number of cells of equal dimension, the maximum cell is determined, and that cell is divided in turn. After subdivision each cell is evaluated to see if the maximum occurs within it. The evaluation can be performed by a numerical integration method such as Simpson's Rule or the Adapted Simpson's Rule. The integrating function incorporates a measure of the investor's relative risk aversion. A number of division/evaluation passes are made until the maximum has been located to within a predetermined margin of error. Systems and media are provided to carry out the above embodiments of the invention. Further aspects of the invention and their advantages can be ascertained from the detailed description set forth below, when taken in conjunction with the drawings, in which like characters illustrate like parts and in which: In the allocation model of the present invention it is assumed that the investor is choosing the allocations of his or her retirement portfolio to maximize his or her utility. In the illustrative embodiment, it is assumed that there are only four different assets to choose from: 1) a nonannuity risk-free asset; 2) a nonannuity risky asset; 3) an immediate fixed annuity; and 4) an immediate variable annuity. The present invention, however, is applicable to scenarios in which there are more kinds of assets available, and is further applicable to situations in which the number of nonannuity assets is not the same as the number of available annuity assets. In another alternative approach to the four-asset scenario described below, the four asset classes illustrated herein can themselves be constructs or portfolios which consist of many other investment instruments whose riskiness and rate of return average to the values shown. Finally, it should be noted that while the illustrated embodiment assigns an equal rate of return for the nonrisky annuity R and the nonrisky non-annuity R, and an equal expected rate of return to a risky annuity X and a risky non-annuity X, this need not be the case. The illustrated embodiment describes the invention using two primary investments (a stock mutual fund and a fixed-income mutual fund) and two immediate annuities based on these. But the invention has application to derivatives as well, such as options, collars and floors. Further, the selected assets to which the allocable wealth is to be distributed may not behave linearly; many immediate variable annuity (IVA) products have downside protection, such that the random return variable associated with the asset is truncated on the bottom. The category matrix below summarizes the random returns from the four possible asset classes shown in the illustrated embodiment.
Each of these four assets is assigned a respective investment weighting factor a -
- w denotes the initial wealth level
- W denotes the final wealth level
- A denotes the relative strength of the utility of consumption.
- D denotes the relative strength of the utility of bequest. The sum of A and D is assumed to be one, so there is only one free variable. Individuals with no utility of bequest will be assumed to have D=0. In other words, the investor chooses A and D according to the amount of wealth the investor wishes to leave to his or her heirs. In the preferred embodiment, it is assumed that A and D≧0 and A+D=1, but A and D can be chosen otherwise.
- p denotes the one period objective probability of survival, which is the probability that is used by insurance companies to price immediate annuities. This factor can be retrieved from an actuary table and varies according to age, gender and ethnic origin, although it can be chosen as a function of less than all of these three factors.
- {overscore (p)} denotes the subjective probability of survival. The subjective probability of survival may not match the objective (annuitant) probability. In other words, a person might believe he or she is healthier (or less healthy) than average. This would impact the expected utility, but not the payout from the annuity, which is based on objective (annuitant) population survival rates.
- {tilde over (X)} denotes the (one plus) random return from a preselected risky asset, e.g., equities, and R denotes the (one plus) risk-free rate of return of a relatively risk-free asset, e.g., U.S. Treasury Bill rate. In the illustrated embodiment, it is assumed that one pair of risky annuity and nonannuity investment vehicles can be chosen to both have a net return {tilde over (X)}, and that another, less-risky pair of annuity and nonannuity investment vehicles can be chosen which have a net return R.
- The expression E[u(α
**1**wR+α**2**w{tilde over (X)}+α**3**wR/p+α**4**w{tilde over (X)}/p)] denotes the utility from the live state, while E[u(α**1**wR+α**2**w{tilde over (X)})] denotes the utility from the dead state. Notice that the annuity terms, which are each divided by the objective probability of survival, do not appear in the dead state. This is because the annuity does not pay out if the investor dies, but the nonannuity assets still do. The ratio of risk free to risky assets is determined as a function of the investor's relative risk aversion, quantified herein as c and Gamma, as will be detailed below. - The function u(.) denotes the standard utility function of end-of-period wealth.
A more general utility function applicable to m annuity assets and n annuity assets can be written as follows: a {overscore (X)} b {overscore (Y)} m=number of nonannuity assets n=number of annuity assets {overscore (X)} In the above equation, within each asset group (annuity or nonannuity), it is assumed that each asset differs from each other asset in risk (measured for example by standard deviation) and return (measured for example as the mean of a distribution of random returns), and that the assets occupy points on an efficient frontier, i.e., as the degree of risk decreases, the degree of return decreases also. The model of the present invention incorporates the following decision factors: (i) Investor's risk tolerance; (ii) Investor's age; (iii) Investor's subjective probability of survival; (iv) Population objective (pricing) probability of survival; (v) Investor's “live” consumption and bequest; (vi) Investor's utility from “live” consumption and bequest; (vii) Expenses and fees of annuity vs. non-annuitized investment instruments (these are simply deductions from the gross returns in order to obtain net returns, in the illustrated four-asset example R and {tilde over (X)}); and (viii) Risk and return characteristics of risky and risk-free assets. In the illustrated embodiment the mean (μ) and standard deviation (σ) have been used to define expected rate of return and risk, and the use of mean and standard deviation in turn assumes the use of a standard log-normal distribution. The probability density function for a log-normal distribution is:
In the illustrated embodiment, the annuity and nonannuity equity-based instruments have a random return X within a distribution defined by two moments: a standard mean variation and a statistical log-normal mean. But other distributions can be used: standard normal, stable and other distributions that have more or less skewness and kurtosis. Similarly, the invention has application to models using serial correlations of returns. For example, a standard normal distribution with no skew can be used. The probability density function for a standard normal distribution is:
Alternatively a logistic distribution with fatter tails than a standard normal distribution can be used. A probability distribution function producing such kurtosis is:
Other distributions can be found in William H. Greene, Econometric Analysis, Prentice Hall, 3rd ed. (1997). In the above expression (1), the ratio of risk-free to risky assets is chosen as a function of an investor-specific risk aversion factor, gamma, and consumption c. But there are literally endless ways of specifying the objective function (utility function) with respect to risk aversion. Here are three other commonly used examples. a) Constant Relative risk aversion utility function (CRRA), where utility is a function of the wealth (W) and γ is the measure of constant relative risk aversion and is always greater than zero.
b) Constant absolute risk aversion utility function (CARA), where utility is a function of the wealth (W) and γ is the measure of an investor's constant absolute risk aversion and is non-negative.
c) Loss Aversion Constant Relative risk aversion Utility Function, where loss aversion is added to the standard CRRA function. W Once the utility function and its inputs have been defined, the next step is to find a maximum of it. Since in the illustrated four-asset embodiment the weights (a Although E[U(W)] is a non-linear function of the three free parameters (α Several methods exist for finding this maximum. The ones discussed herein involve (a) defining a search space, (b) dividing the search space into intervals, (c) evaluating the utility function within the interval by integrating it, (d) choosing which interval appears to have the maximum within it, and repeating steps (b)–(d) by subdividing the previously selected interval, until the maximum has been located to an acceptable level of accuracy. A preferred search method uses the Golden Mean or Golden Section. According to this method, the optimal value of the objective utility function is assumed to have been bracketed in an interval [a,b]. In the present invention, the interval is defined by the variations of allocations across all permissible asset classes. In the four-asset class example this will vary from 100% nonannuity cash (or other relatively risk-free nonannuity investment), 99% cash+1% nonannuity equity (or other relatively risky nonannuity investment), . . . , to 100% variable income annuity (VIA). One then evaluates the utility function at an intermediate point x and obtains a new smaller bracketing interval, either [a,x], or [x,b]. This process continues until the bracketing interval is acceptably small. It is critical to choose x to be the optimal point between [a,b] so that the decrease in the interval length is maximized when the function is as uncooperative as it can be, i.e., when the luck of the draw forces you to take the bigger bisected segment. According to the golden section, the optimal bracketing interval point x between [a,b] should be a fractional distance 0.38197 from one end (say a), and 0.61803 from the other end (say b). This process can be expanded to a multi-dimension optimization. For a general discussion on optimization methods, see William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling, An alternative search method according to the present invention uses the so-called Method of Feasible Directions, which will be explained herein in further detail below. In optimizing the four-asset objective utility function by the Method of Feasible Directions, the following steps are performed: -
- 1) Choose an initial feasible point B=(b
_{1}, b_{2}, b_{3}) (i.e. b_{1}≧0, b_{2}≧0, b_{3}≧0 and b_{1}+b_{2}+b_{3}≦1). - 2) Solve the following linear program for (d
_{1}, d_{2}, d_{3}, d_{4}):- Maximize: z=d
**4** - Subject to:
- −∇E[U(W)](B)·D+d
**4**≦**0** - −d
**1**≦b**1** - −d
**2**≦b**2** - −d
**3**≦b**3** - d
**1**+d**2**+d**3**≦1−(b**1**+b**2**+b**3**),
- −∇E[U(W)](B)·D+d
- Maximize: z=d
- where ∇E[U(W)](B) denotes the gradient of E[U(W)] evaluated at the point B, D=(d
_{1}, d_{2}, d_{3}), and “·” denotes the dot product of two three-vectors. If d_{4}=0, then X*=B is the point at which E[U(W)] is maximum; if not, then - 3) Determine a nonnegative value for λ that minimizes the single-variable strictly concave function E[U(W)] (B+λD), while keeping (B+λD) feasible (that is, b
_{1}+λd_{1}>0, b_{2}+λd_{2}≧0, b_{3}+λd_{3}≧0, and (b_{1}+λd_{1})+(b_{2}+λd_{2})+(b_{3}+λd_{3})≦1). Let λ* denote this optimal value of λ. - 4) Set B=B+λ*D, and go back to step (2).
- 1) Choose an initial feasible point B=(b
Step (2) requires (after restating the linear program in “standard” form) the use of the simplex algorithm. The second, third and fourth constraints will acquire slack variables when put in standard form, and the fifth constraint will acquire a surplus variable, and hence also an auxiliary (a.k.a. artificial) variable. Depending on the components of the gradient ∇E[U(W)](B), the first constraint may acquire either a slack variable or a surplus (and hence also an auxiliary) variable. In any event, the presence of at least one auxiliary variable will require the use of the Two-Phase Simplex Method; Phase I determines an initial feasible point of the linear program if it exists (in our case it does), and Phase II determines the optimal point of the linear program. Technically, step (3) cannot be implemented as written above, since the condition “d Step (3) will require an additional method such as the Fibonacci Search or Golden Mean Search (GMS) algorithms for maximizing a function of a single variable. According to a presently preferred MFD embodiment, the Golden Mean Search algorithm is used because it is easier to code compared with the Fibonacci Search algorithm and almost as fast. However, one of ordinary skill in the art will appreciate that other algorithms for maximizing a function of a single variable may readily be substituted. The number of iterations of the algorithm depends on the desired proximity of the calculated optimal point to the true optimal point. Another search method which may be employed is a “basic” search, in which the search space is divided into a predetermined number of intervals, the function evaluated in each of them, one of the intervals then being chosen based on the maximum residing within it, the chosen interval then being subdivided into the predetermined number of subintervals, and repeating these steps until the maximum has been localized to within a predetermined degree of accuracy. There are a number of ways to perform the required numerical integrations of E[U(W)] and ∇E[U(W)]; two are the Simpson's Composite Rule and the Adapted Simpson's Rule. For either, since the integrals in E[U(W)] and ∇E[U(W)] are over the unbounded interval [0,∞] we must break the integral into one over [0,1] and one over [1,∞]. The variable of integration x in the [1, ∞] interval is changed to y=1/x, therefore changing the interval to [0,1], and the two integrals over [0,1] are evaluated together. In certain cases, the numerical integration is not required. One such case is where the portfolio returns is log-normally distributed and the utility function is Constant Relative Risk Aversion. In this case, the moments of log-normal distribution are available numerically. In one embodiment, the Adapted Simpson's Composite Rule is used because this technique makes it possible to impose a maximum error size by keeping track of the variation of the function being integrated, and using finer intervals where the function varies more, and larger intervals where the function varies less. According to the illustrated embodiment, Simpson's Composite Rule is used with a “basic” search, by which are established 40 equally spaced subintervals over the interval [0,1]. For the interval [1, ∞], the feasible region α At the next stage, the part of the box [A The first step MeanX and StdDevX are the mean and standard deviation of a random distribution of returns (either lognormal or other distributions as previously discussed) from an investment in a relatively risky asset such as equities. This can be based on the realized, historical returns for these assets, or any other model or estimate known to those of ordinary skill in the art. Finally, c and Gamma are the coefficient of non-relative risk aversion and the coefficient of risk aversion, respectively. In step In steps The number of increments impacts the accuracy of the numerical integration, and that accuracy may be increased by increasing the number of increments. However, there is a tradeoff between the number of increments and the speed of execution of the calculation. In steps Next, the first pass using Simpson's Composite Rule is performed in step Details of the first pass The flowchart of The integrands differ depending on whether Gamma< >1 or Gamma=1. According to the preferred embodiment, an if-then statement If Gamma, the coefficient of risk aversion, is not equal to 1 then the integration is performed using first integrands One of ordinary skill in the art will appreciate that in the for-next loop Details of the second pass The flowchart of As shown in step The variable f (calculated at The variable Aint The variable Aint The variable Dint The function Dint The function atn(x) (see step As shown in One will recall that the B In the expressions “w” is set to 1 since the optimal asset mix does not depend on the wealth level w. “R” is 1 plus the risk-free rate of return (e.g., 1.03). “p” is the objective probability of survival. “pbar” ({overscore (p)}) is the subjective probability of survival. “h” is the step size in the numerical integration of E[U(W)]. “Gamma” is the coefficient of relative risk aversion. Referring back to In the steps After the first calculation of EU, MAX is set to EU−1. Referring back to The second pass again uses three nested For . . . Next loops It should be appreciated that the outermost loop cannot start at B Like the first pass ( As shown in step Next, “EA”, the expectation in the “A” part of E[U(W)], “ED”, the expectation in the “D” part of E[U(W)], and “EU”, the expectation of E[U(W)] are calculated in steps If a new maximal value for EU is determined in step Referring once again to As will be explained below, the optimal allocation weights may be output in step Preferably the found values A In many cases, there will be some funds or investments of the investor which may not be reallocated—but it is still important that the overall financial wealth of the investor be correctly balanced among annuity, nonannuity, risky and relatively risk-free assets. This may happen, for example in a situation in which a financial planner has control over only a portion of the investor's financial investments. Such a case is illustrated in the flowchart of At step In an alternative embodiment, the investment weighting factors can be used to construct better products, typically by adding annuity assets to existing investing company products. In another alternative, the investment weighting factors can be used as a tool by a financial planner to reallocate the assets of an investor after making a comparison to present assets. Such a use for the investment weighting factors is shown in At step A representative system suitable for carrying out the invention is illustrated in Referring to The schematic representation of bus Computer In alternative embodiments of the invention, one or more components of the invention's logic may be “hard-wired” into the ROM In a typical architecture, a computer program suitable for carrying out the invention will be stored on a mass storage device The computer As previously described, the asset allocation weights a The financial advisor computer The computer To understand the predictions of the model of the present invention, let us look at four different cases so that we can see the effect of changing parameters on the optimal allocation. Each case uses the same capital market assumptions. For the sake of example assume that the return from the risk-free (T-bills) asset class is 5% per annum with no volatility. Likewise assume that the return from the risky asset is log-normally distributed with a mean value of 10% and a standard deviation of 20%. This implies a risk premium of 7%, which is in line with historical estimates. Many people might feel they are less (or more) healthy than the numbers indicated by the Individual Annuity Mortality 2000 basic table of The utility preferences for the example will be taken from within the Constant Relative Risk Aversion (CRRA) family, with a CRRA coefficient of γ. CRRA coefficient γ typically has a value in the range of 1 to 6, although it can be as high as 10 or 20. Gamma can be obtained as a result of the answers of the investor to a questionnaire. Each of the below listed cases will assume that the individual is a 60-year-old male who would like to allocate his portfolio across the four asset classes (1) risk-free asset; 2) risky asset; 3) immediate fixed annuity; and 4) immediate variable annuity). Finally, each case will assume a 20-year horizon as representing the one-period. In other words the individual intends to liquidate the immediate annuities, consume, and re-allocate (i.e. rebalance) assets only after 20 years. It should be understood that these assumptions can be easily modified to accommodate other utility functions, asset return distributions, mortality probabilities, and horizons. Moreover, the initial wealth level does not have any impact on the allocations for the one-period model. Case #1: Total Altruism and Complete Bequest Motives In this case, we assume the investor's utility is derived entirely from bequest. In other words his utility of bequest is assumed to be one and his utility of consumption is zero, that is, A=0 and D=1. The objective probability of survival is 65% (roughly equal to the survival probability of a 60-year-old male in the next 20 years) and the subjective probability is the same 65%. Using these input parameters in the model described in Eqn. (1) above, the optimal allocations to the assets across various relative risk aversion levels are as presented in As shown in Case #2: No Bequest Motives This case maintains the same age (gender), survival probability and time horizon as the previous case, but simply reduces the strength of bequest from D=1 to D=0. In other words, 100% of the utility weight is placed on ‘live’ consumption. The optimal allocations to the assets across various risk aversion levels are presented in As shown in Case #3: 20% Bequest Motives and 80% Consumption Motives This case maintains the same age (gender), survival probability and time horizon as the previous case, but changes the strength of bequest from D=0, to a more realistic D=0.2. In other words, 80% of the utility weight is placed on ‘live’ consumption. The optimal allocations to the assets across various risk aversion levels are presented in There are several interesting results in the allocation shown in Second, the allocation between risky (both variable IA and equity) and risk-free (cash and fixed IA) is almost identical to that in Case #1 and Case #2 at comparable risk aversion levels. This indicates that the changes in the investor's bequest vs. consumption motive do not significantly impact the investor's behavior regarding risk. The optimal allocation between risky and risk-free assets is determined by the investor's risk tolerance. Third, the allocation to annuities decreases as the investor's risk aversion increases. In other words, more risk averse investors will avoid immediate life annuities. This makes intuitive sense, since the investor could get little or no utility from immediate annuity investments if he dies shortly after the purchase. With traditional investments, there will be some left for their heirs. It seems that higher aversion to risk increases the implicit weight on the utility of bequest. For an investor with a risk aversion level of 2, the optimal allocation is 22% cash (“cash” being used herein as shorthand for an efficient investment vehicle of very low risk), 38% equity, 14% FIA, and 26% variable IA. Case #4: Lower Subjective Survival Probability In the fourth case, we maintain the same parameters as the third case but change the subjective survival rate ({overscore (p)}) to be smaller than the objective survival rate (p). We assume ({overscore (p)})=0.5 and p=0.65. All other parameters are left the same. The implication is simple. The investor believes that he/she has a lower than average probability of survival. Intuitively, such an investor would allocate less to immediate annuities, since he/she is expecting a higher mortality probability than the average. See In The above four cases have assumed a single male as the investor. This can be changed to single female or a married couple simply by replacing the probability of survival to parameters that are accurate for a single female or for the joint probability of survival for a couple. In summary, a method and system for performing optimal asset allocation in the presence of fixed and variable immediate life annuities has been shown and described. While preferred embodiments of the present invention have been described in the above detailed description, and illustrated in the drawings, the invention is not limited thereto but only by the scope and spirit of the appended claims. Patent Citations
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